🔁 What is an endomorphism in general?
In mathematics, an endomorphism is a function from a space to itself that preserves the structure of that space.
- For a group, it’s a group homomorphism \(f: G \rightarrow G\)
- For a vector space, it’s a linear map \(f: V \rightarrow V\)
- For an elliptic curve, it’s a map \(\phi: E \rightarrow E\) that respects the curve’s group law.
notes:
🧮 Homomorphism vs. Endomorphism
| Term | Definition | Example |
|---|---|---|
| Homomorphism | A structure-preserving map between two algebraic structures (like groups or rings). | \(f: G \rightarrow H\), where \(f(a \cdot b) = f(a) \cdot f(b)\) |
| Endomorphism | A homomorphism from a structure to itself. | \(\phi: G \rightarrow G\) |
✅ Key Differences
1. Direction of the map
- Homomorphism: From one structure to another (could be different!)
- f:A→Bf: A \rightarrow Bf:A→B
- Endomorphism: From a structure to itself
- f:A→Af: A \rightarrow Af:A→A
2. Scope
Not all homomorphisms are endomorphisms — only the ones mapping back to the same object.
All endomorphisms are homomorphisms.
4. In elliptic curves
- Homomorphism: Could be a map from one curve to another (e.g., f:E1→E2f: E_1 \to E_2f:E1→E2)
- Endomorphism: A map from a curve to itself (e.g., ϕ:E→E\phi: E \to Eϕ:E→E) that respects point addition
- A homomorphism is a map between two structures of the same type (same algebraic category) that preserves the relevant operations.
- Group homomorphism: between two groups
- Ring homomorphism: between two rings
- Field homomorphism: between fields
🔁 1. What is an Endomorphism?
An endomorphism is a function \(\phi: E \rightarrow E) that takes points on an elliptic curve \(E\) and maps them to other points on the same curve, in a way that preserves the group operation: \(ϕ(P+Q)=ϕ(P)+ϕ(Q)\phi(P + Q) = \phi(P) + \phi(Q)ϕ(P+Q)=ϕ(P)+ϕ(Q)\)
This makes ϕ\phiϕ a group homomorphism from the curve to itself.
There are always trivial endomorphisms:
- Identity: ϕ(P)=P\phi(P) = Pϕ(P)=P
- Doubling: ϕ(P)=2P\phi(P) = 2Pϕ(P)=2P
- Tripling: ϕ(P)=3P\phi(P) = 3Pϕ(P)=3P, etc.
But the interesting and useful ones are the non-trivial algebraic endomorphisms that come from special symmetries of the curve.